The image via β2\beta_2 of the nn-th Stiefel-Whitney mapwn∈H(X,Bnℤ/2ℤ)w_n\in \mathbf{H}(X,\mathbf{B}^n\mathbb{Z}/2\mathbb{Z}) in H(X,Bn+1ℤ)\mathbf{H}(X,\mathbf{B}^{n+1}\mathbb{Z}) is called the (n+1)(n+1)st integral Stiefel-Whithey map and is denoted by Wn+1W_{n+1}.

One usually uses the same symbol to denote the image of this characteristic map in cohomology (on connected components ) of Wn+1W_{n+1} in Hn+1(X;ℤ)=π0H(X,Bn+1ℤ)H^{n+1}(X;\mathbb{Z})=\pi_0\mathbf{H}(X,\mathbf{B}^{n+1}\mathbb{Z}), and calls this the (n+1)(n+1)-th integral Stiefel-Whitney class.

Since H2(X;ℤ)H^2(X;\mathbb{Z}) classifies isomorphism classes of U(1)U(1)-principal bundles over XX and W3(TX)W_3(T X) is the obstruction to the existence of a spin^c structure on XX, we see that XX has a spincspin^c structure if and only if there exists a principal U(1)U(1)-bundle on XX “killing” the second Stiefel-Whitney class of XX.

In particular, when w2(TX)w_2(T X) is killed by the trivial U(1)U(1)-bundle, i.e., when w2(TX)=0w_2(T X)=0, then XX has a spin structure.